Sparse sampling of signal innovations

ABSTRACT

Signals, including signals from outside of the subspace of bandlimited signals associated with the Shannon theorem, are acquired while still providing an acceptable reconstruction. In some aspects a denoising process is used in conjunction with sparse sampling techniques. For example, a denoising process utilizing a Cadzow algorithm may be used to reduce the amount of noise associated with sampled information. In some aspects the denoising process may be iterative such that the denoising process is repeated until the samples are denoised to a sufficient degree. In some aspects, the denoising process converts a set of received samples into another set corresponding to a signal with a Finite Rate of Innovation (FRI), or to an approximation of such a signal. The disclosure relates in some aspects to combination of a denoising process with annihilating filter methods to retrieve information from a noisy, sparse sampled signal. The disclosure relates in some aspects to determining a sampling kernel to be used to sample the signal based on noise associated with the signal. The disclosure relates in some aspects to determining the number of samples to obtain from a signal over a period of time based on noise associated with the signal. The disclosure relates in some aspects to determining the finite number of innovations of a received signal.

CLAIM OF PRIORITY UNDER 35 U.S.C. §119

This application claims the benefit of and priority to commonly ownedU.S. Provisional Patent Application No. 61/024,490, filed Jan. 29, 2008,and assigned Attorney Docket No. 080467P1, and U.S. Provisional PatentApplication No. 61/056,565, filed May 28, 2008, and assigned AttorneyDocket No. 080467P2, the disclosure of each of which is herebyincorporated by reference herein.

CROSS-REFERENCE TO RELATED APPLICATION

This application is related to concurrently filed and commonly ownedU.S. Patent Application entitled “SPARSE SAMPLING OF SIGNALINNOVATIONS,” and assigned Attorney Docket No. 080467U2, the disclosureof which is hereby incorporated by reference herein.

BACKGROUND

1. Field

This application relates generally to signal processing and morespecifically, but not exclusively, to wireless telecommunication, signalacquisition and reconstruction.

2. Introduction

Signal acquisition and reconstruction is at the heart of signalprocessing, and sampling theorems provide the bridge between continuoustime phenomena and discrete-time representations of such phenomena. Awell-known sampling theorem is often attributed to Shannon, and gives asufficient condition, namely bandlimitedness, for an exact sampling andinterpolation formula. The minimal sampling rate, at twice the bandwidthof the analog signal, is typically referred to as the Nyquist rate.

The Shannon case is a particular example, where any signal from thesubspace of bandlimited signals denoted by BL, can be acquired throughsampling and perfectly interpolated from the samples. Using the sinckernel, or ideal lowpass filter, non-bandlimited signals will beprojected onto the subspace BL.

International Patent Application WO 02/078197, which is herebyincorporated by reference, develops sampling schemes for a larger classof non-bandlimited signals, such as streams of Diracs, non-uniformsplines and piecewise polynomials. A common feature of these signals isthat they have a parametric representation with a finite number ofdegrees of freedom (or a number which is finite in each period), and canbe perfectly reconstructed from a finite set of samples.

SUMMARY

A summary of sample aspects of the disclosure follows. It should beunderstood that any reference to the term aspects herein may refer toone or more aspects of the disclosure.

The disclosure relates in some aspects to acquiring signals, includingsignals from outside of the subspace of bandlimited signals associatedwith the Shannon theorem, while still providing an acceptablereconstruction. In some aspects this may involve taking advantage ofsome sort of sparsity in the signal being reconstructed. Through the useof sparse sampling at a rate characterized by how sparse signalcomponents are per unit of time, the Nyquist constraints may be avoidedwhile accurately sampling and reconstructing signals. In some aspects,sampling is performed at the rate of innovation of the signal.

The disclosure relates in some aspects to using a denoising process inconjunction with sparse sampling techniques. For example, a denoisingprocess utilizing a Cadzow algorithm may be used to reduce the amount ofnoise associated with sampled information. In some aspects the denoisingprocess may be iterative such that the denoising process is repeateduntil the samples are denoised to a sufficient degree. This denoisingprocess is especially useful when applied to signal sampled atsub-Nyquist rates, but can also be used with bandlimited signals sampledat rates higher than Nyquist.

In some aspects, the denoising process converts a set of receivedsamples into another set corresponding to a signal with a Finite Rate ofInnovation (FRI), or to an approximation of such a signal. The denoisingprocess thus removes or reduces at least one component of the noise,i.e., the component that makes the signal have a non Finite Rate ofInnovation. In one aspect, the parameters (amplitude (weight) andlocation (phase, shift)) of the set of samples delivered by thedenoising process may still be noisy, but the signal-to-noise ratio isimproved by this process.

The output of an infinite number of iterations of Cadzow is thus notonly “denoised” but is actually a signal with a Finite Rate ofInnovation, very unlike the noisy signal. By using a sufficient numberof Cadzow iterations, the output may be an FRI signal (e.g., on theorder of 1e-10). Such an FRI signal may equivalently be represented,either by a sequence of uniform samples, or by the set of parameters(innovations) that provide these samples. The switch between the tworepresentations is performed by using an annihilation filter technique.In the disclosed approach this technique may be reduced to be theinverse operation of the sampling operation.

For the retrieval of a parametric model, a subspace technique can beused. In particular, the matrix of coefficients derived from the sampleshas a structure and rank condition which can be taken advantage of. Asan example, the matrix of Fourier coefficients is Toeplitz and has rankK when there are K Diracs in the time domain signal. Thus, a singularvalue decomposition can be used to get a rank K approximation to thenoisy matrix, which is a subspace approximation.

The disclosure relates in some aspects to a method where a signal with anon Finite Rate of Innovation, for example a noisy signal, can beacquired through sampling and projected onto the subspace of signalswith a Finite Rate of Innovation, allowing a perfect or at leastimproved interpolation from the samples.

The disclosure relates in some aspects to determining the number ofsamples to obtain from a signal over a period of time based on noiseassociated with the signal. For example, the number of samples may beselected based on the known or assumed signal-to-noise ratio, and/or onthe desired accuracy of the reconstructed signal.

The disclosure relates in some aspects to determining (e.g., defining) asampling kernel to be used to sample the signal based on noiseassociated with the signal. For example, the amount and type of noise ina signal may affect the bandwidth of a sampling kernel and/or the typeof a sampling kernel to be used to sample the signal. The bandwidth ofthe sampling kernel, in turn, may affect the number of samples toacquire.

The disclosure relates in some aspects to determining the finite numberof innovations of a received signal. For example, a number ofinnovations of a signal may be determined based on at least one rank ofat least one matrix that is defined based on the received signal. Insome aspects such a matrix may be associated with an annihilationfilter.

The disclosure also relates in some aspects to combination of adenoising process with annihilating filter methods in order to retrieveinformation from a noisy, sparse sampled signal.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other sample aspects of the disclosure will be described inthe detailed description and the appended claims that follow, and in theaccompanying drawings, wherein:

FIG. 1 is a simplified block diagram of several sample aspects of acommunication receiver, with schematic indications of potential noiseperturbations in the analog part and in the digital part.

FIG. 2 is a simplified block diagram of several sample aspects of acommunication receiver, with schematic indications of potential noiseperturbations in the analog part and in the digital part;

FIG. 3 is a simplified block diagram of several sample aspects of acommunication system;

FIG. 4 is a simplified block diagram of several sample aspects ofcommunication components; and

FIGS. 5A and 5B are simplified block diagrams of several sample aspectsof apparatuses configured to provide signal acquisition andreconstruction as taught herein.

In accordance with common practice the various features illustrated inthe drawings may not be drawn to scale. Accordingly, the dimensions ofthe various features may be arbitrarily expanded or reduced for clarity.In addition, some of the drawings may be simplified for clarity. Thus,the drawings may not depict all of the components of a given apparatus(e.g., device) or method. Finally, like reference numerals may be usedto denote like features throughout the specification and figures.

DETAILED DESCRIPTION

Various aspects of the disclosure are described below. It should beapparent that the teachings herein may be embodied in a wide variety offorms and that any specific structure, function, or both being disclosedherein is merely representative. Based on the teachings herein oneskilled in the art should appreciate that an aspect disclosed herein maybe implemented independently of any other aspects and that two or moreof these aspects may be combined in various ways. For example, anapparatus may be implemented or a method may be practiced using anynumber of the aspects set forth herein. In addition, such an apparatusmay be implemented or such a method may be practiced using otherstructure, functionality, or structure and functionality in addition toor other than one or more of the aspects set forth herein. Furthermore,an aspect may comprise at least one element of a claim.

Referring initially to FIGS. 1 and 2, the sampling and reconstructionscheme of the present disclosure will be described in a theoreticalmanner. FIG. 1 relates to a system that may be employed for a noiselesscase and, potentially, for a substantially noiseless case (e.g., wherethe effect of noise may be ignored or will be automaticallycompensated). The system comprises a transmission channel which possiblyadds an analog noise ε(t) to a transmitted signal (the “another” signal)x(t). The analog signal y(t) transmitted over the transmission channelis received by a suitable receiver 305 in an apparatus 302. The receiver305 may comprise for example an antenna and a radiofrequency part (notshown). In some aspects, the apparatus 302 may comprise a sampler thatsamples a received signal y(t) using a sampling kernel. For example, inFIG. 1 the received signal y(t) is filtered by a suitable samplingkernel φ(t), for example a sinc filter, with a suitable bandwidth. Thefiltered signal is then sampled at a sample frequency 1/T, and possiblyquantized. It should be appreciated that in some aspects the sampler maycomprise at least a portion of the sampling filter. In some aspects atleast one of: the receiver 305, the sampling kernel (e.g., a filterbased on the sampling kernel), or the sampler may be implemented in acircuit in the apparatus 302.

The quantization process, error corrections or various perturbations inthe receiver 305 or elsewhere in the apparatus 302 may add a digitalnoise ε(n) to the sampled signal y(n) (which may be referred to hereinas y_(n)). Here, it should be appreciated that the “addition block” issimply intended to illustrate, for example, that at some point digitalnoise ε(n) may be imparted on the sampled signal y(n) whereby the inputto an estimator 303 may be a composite signal comprising y(n) and ε(n).In the example illustrated in FIG. 1, the analog noise ε(t) and thedigital noise ε(n) are both null or at least low compared to the signal,so that a denoising process may not be needed, as will be described.

In order to reconstruct the original signal x(t), the sampled signaly(n) with the additional digital noise ε(n) is then converted in thefrequency domain, using for example a Fast Fourier Transformation (FFT)or another suitable transformation. An annihilating filter, as part ofthe estimator 303, is then computed in order to retrieve shifts t_(k),and then weights x_(k), by resolving a linear system in the estimator.The original signal x(t) may then be accurately reconstructed from theset of retrieved innovations t_(k), x_(k) (e.g., from estimates of theshifts and weights). In some cases, a complete reconstruction of theanother signal x(t) (which may be referred to herein as x_(t)) is notrequired and it may be sufficient to retrieve some information(“innovations”) corresponding to this signal, for example only theshifts t_(k), only the weights x_(k), and/or other information relatingto x(t).

FIG. 2 relates to a system that may be employed for the case of noisysignals. Comparing with FIG. 1, in this system the estimator 303 alsocomprises a denoiser for replacing the sampled signal y(n) (or the FFTtransform of this signal) by an approximation with no or less noise,i.e. a “denoised” sequence of samples (y′_(n)). In this context,“denoised” means that at least a part of the noise has been removed orreduced. A denoiser is thus an apparatus, part or software module thatconverts a noisy signal into another signal from which a better, lessnoisy estimation of the required information can be computed.

In the example of FIG. 2, the annihilating filter and the linear systemare shown as being implemented in a processor component 306. The weightsx_(k) and shifts t_(k), or other information relating to the anothersignal x(t), are thus retrieved from the denoised signal y′(n) (whichmay be referred to herein as y′_(n)) using a similar annihilating filterand by solving a similar linear system. The denoising may be applied tothe samples in the time domain or, as illustrated, in the frequencydomain after a suitable transformation such as a Fast FourierTransformation (FFT).

In the aspects of FIGS. 1 and 2, the sampling rate 1/T used for samplingthe (filtered) received signal y(t) (which may be referred to herein asy_(t)) may be sub-Nyquist, i.e. lower than the minimal sampling rategiven by the Shannon theorem applied to the another signal x(t) and/orto the received signal y(t), while still allowing an accurate or (if thesignal-to-noise ratio is sufficient) even perfect reconstruction of theanother signal x(t). One of the options associated with the samplingrate is to choose it higher than the finite rate of innovation ρ of theanother signal x(t). The method thus even applies to a non bandlimitedsignal, such as a sequence of Diracs, a rectangular signal, a piecewisepolynomial signal, any signal with a finite duration and finite numberof innovations during this duration, any periodic signal with a finitenumber of innovations during each period, etc., or to a signal y(t) thatincludes a noise component making it not bandlimited.

The sampling rate 1/T used for sampling the received signal y(t) thusdepends mainly on the rate of innovation of the another signal x(t).This rate of innovation may be previously known, estimated, assumed, orretrieved from the signal itself, for example in a receiver. Thesampling rate also depends on the level of noise ε(t) and ε(n), forexample on the signal-to-noise ratio, which may also be previouslyknown, estimated, assumed, or retrieved from the received signal. Inaddition, the determination of the sampling rate depends on the desiredlevel of accuracy of the information one wants to retrieve. In anaspect, the determination of the sampling rate is based on a comparisonbetween the signal-to-noise ratio of the reconstructed signal with anassumption of an expected signal to noise ratio. Thus, in some aspectsthe sampling rate may be defined before the receipt of the anothersignal x(t) by the apparatus 302. In some aspects the apparatus 302(e.g., one or more of the estimator, the receiver, the sampler, or adedicated rate determination circuit) may determine the sampling ratebased on an estimate of the noise (e.g., estimated signal to noiseratio), a desired level of accuracy, a comparison of signal to noiseratios, or the rate of innovation (e.g., based on K which may bedetermined as discussed below).

The sampling kernel φ(t), in particular the bandwidth and/or functionused as a sampling kernel, may also depend on the rate of innovation ofthe another signal x(t) and/or on noise, or on the signal-to-noiseratio, or on the type of noise, or more generally on an estimate of theanalog noise ε(t) and/or digital noise ε(n). Thus, in some aspects theapparatus 302 (e.g., one or more of the estimator, the receiver, thesampler, or a dedicated kernel defining circuit) may define the kernel(e.g., by adjusting the bandwidth of an adjustable kernel filter) basedon the noise (e.g., an estimate of noise) or the rate of innovation(e.g., based on K which may be determined as discussed below).

Sample implementations of this scheme will be described in conjunctionwith FIG. 3. FIG. 3 illustrates a communication system with atransmitter 301 that sends an analog signal x(t) over a noisycommunication channel, for example a wireless or wired communicationchannel. An apparatus 302 receives a signal y(t) based on x(t) and onnoise ε(t) added by the communication channel (e.g., the signal y(t) isa composite signal comprising x(t) and ε(t)), and retrieves informationcorresponding to x(t) by sampling y(t) at a sub-Nyquist rate, possiblydenoising the sampled signal, and applying suitable annihilating filterand linear system estimation methods as described below.

Signals with Finite Rate of Innovation

Using the sinc kernel (defined as sinc t=sin πt/πt), a signal x(t)bandlimited to [−B/2,B/2] may be expressed as set forth in Equation 1:

$\begin{matrix}{{x(t)} = {\sum\limits_{k \in Z}{x_{k}\sin \; {c\left( {{Bt} - k} \right)}}}} & (1)\end{matrix}$

where x_(k)=<B sinc(Bt−k), x(t)>=x(k/B).

Alternatively, it may be said that the signal x(t) has B degrees offreedom per second, since x(t) is exactly defined by a sequence of realnumbers {x_(k)}_(k∈Z), spaced T=1/B seconds apart. This may be referredto as the rate of innovation of the bandlimited process, denoted by ρ,and equal to B.

A generalization of the space of bandlimited signals is the space ofshift-invariant signals. Given a basis function φ(t) that is orthogonalto its shifts by multiples of T, or <φ(t−kT),φ(t−k′T)>=δ_(k-k′), thespace of functions obtained by replacing sinc with φ in (1) defines ashift-invariant space S. For such functions, the rate of innovation isagain equal to ρ=1/T.

For a generic sparse source, such as a Poisson process, which is a setof Dirac pulses, Σ_(k∈Z)δ(t−t_(k)), where t_(k)-t_(k−1) is exponentiallydistributed with p.d.f.λe^(−λt). Here, the innovations are the set ofpositions (or shifts) {t_(k)}_(k∈Z). Thus, the rate of innovation is theaverage number of Diracs per unit of time: ρ=lim_(T→∞) C_(T)/T, whereC_(T) is the number of Diracs in the interval [−T/2, T/2]. Thisparallels the notion of information rate of a source based on theaverage entropy per unit of time introduced by Shannon. In the Poissoncase with decay rate λ, the average delay between two Diracs is 1/λ;thus, the rate of innovation ρ is equal to λ.

A generalization involves weighted Diracs, as set forth in Equation 2:

$\begin{matrix}{{x(t)} = {\sum\limits_{k \in Z}{x_{k}{\delta \left( {t - t_{k}} \right)}}}} & (2)\end{matrix}$

By similar arguments, ρ=2λ in this case, since both positions (shifts)and weights are degrees of freedom.

A sampling theorem for the type of sparsely sampled signals is describedbelow where ρ samples per unit of time are acquired, which perfectlydescribe the original signal, thereby allowing a perfect reconstructionof the original signal, as with the Shannon sampling procedure but withless samples or a lower sampling rate.

According to one aspect, the sampling rate should be (at least) ρ, therate of innovation of the original signal. To show that is sufficientcan be done in a number of cases of interest. The archetypal sparsesignal is the sum of Diracs, observed through a suitable samplingkernel. In this case, sampling theorems at the rate of innovation can beproven. Beyond the question of a representation theorem, efficientcomputational procedures, showing the practicality of the approach aredescribed. Next, the question of robustness to noise and optimalestimation procedures under these conditions is addressed. Here,algorithms to estimate sparse signals in noise that achieve performanceclose to optimal will be described. This may be done by computingCramer-Rao bounds that indicate the best performance of an unbiasedestimation of the innovation parameters. Note that, when theSignal-to-Noise ratio is poor, the algorithms are iterative, and thustrade computational complexity for estimation performance. Forconvenience, Table 1 sets forth several notations that are used herein.

TABLE 1 Symbol Meaning x(t), τ, {circumflex over (x)}_(m) τ-periodicFinite Rate of Innovation signal and its Fourier coefficients K, t_(k),x_(k) and ρ${{{Innovation}\mspace{14mu} {parameters}\text{:}\mspace{14mu} {x(t)}} = {\sum\limits_{k = 1}^{K}{x_{k}{\delta \left( {t - t_{k}} \right)}}}},{{{for}\mspace{14mu} t} \in \left\lbrack {0,\tau} \right\rbrack}$and rate of innovation of the signal: ρ = 2K/τ φ(t), B “Anti-aliasing”filter, prior to sampling: typically φ (t) = sinc Bt Note: B x τ isrestricted to be an odd integer y_(n), ŷ_(m, N, T) (noisy) samples{y_(n)}_(n=1, 2, . . . , N) of (φ * x)(t) at multiples of T = τ/N (seeEquation 15) and its DFT coefficients ŷ_(m) A, L rectangularannihilation matrix with L + 1 columns (see Equation 13) H(z), h_(k) andH Annihilating filter: z-transform, impulse response and vectorrepresentation

Sampling Signals at Their Rate of Innovation

A τ-periodic stream of K Diracs δ(t) with amplitudes x_(k) located attimes t_(k) ∈ [0, τ] may be represented as:

$\begin{matrix}{{x(t)} = {\sum\limits_{k = 1}^{K}{\sum\limits_{k^{\prime} \in Z}{x_{k}{\delta \left( {t - t_{k} - {k^{\prime}\tau}} \right)}}}}} & (3)\end{matrix}$

This signal x(t) is convolved with a sinc-window of bandwidth B, whereBτ is an odd integer, and is uniformly sampled with sampling periodT=τ/N. We want to retrieve the innovations x_(k) and t_(k) in theoriginal signal from the n=1, 2, . . . ,N measurements:

$\begin{matrix}{{y_{n} = {{\langle{{x(t)},{\sin \; {c\left( {B\left( {{nT} - t} \right)} \right)}}}\rangle} = {\sum\limits_{k = 1}^{K}{x_{k}{\phi \left( {{nT} - t_{k}} \right)}}}}}{where}} & (4) \\{{\phi (t)} = {{\sum\limits_{k^{\prime} \in Z}{\sin \; {c\left( {B\left( {t - {k^{\prime}\tau}} \right)} \right)}}} = \frac{\sin \left( {\pi \; {Bt}} \right)}{B\; \tau \; {\sin \left( {\pi \; {t/\tau}} \right)}}}} & (5)\end{matrix}$

is the τ-periodic sinc function or Dirichlet kernel. Here, x(t) has arate of innovation ρ=2K/τ. During each period τ, the weight x_(k) andshifts t_(k) each take K values.

It is desirable to provide a sampling scheme that is able to retrievethe innovations of x(t) by operating at a sampling rate that is as closeas possible to ρ.

Since x(t) is periodic, we can use the Fourier series to represent it.The Fourier series coefficients of x(t) are thus:

$\begin{matrix}{{{{x(t)} = {\sum\limits_{m \in Z}{{\hat{x}}_{m}^{{j2\pi}\; {{mt}/\tau}}}}},{where}}\mspace{11mu} \; {{\hat{x}}_{m} = {\frac{1}{\tau}{\sum\limits_{k = 1}^{K}{x_{k}\underset{\underset{u_{k}^{m}}{}}{^{{- {j2\pi}}\; {{mt}_{k}/\tau}}}}}}}} & (6)\end{matrix}$

In this noiseless case, the signal x(t) is completely determined by theknowledge of the K amplitudes (weights) x_(k) and the K locations(shifts) t_(k), or equivalently, by the knowledge of u_(k). Byconsidering 2K contiguous values of {circumflex over (x)}_(m) inEquation 6, a system of 2K equations in 2K unknowns may be constructedthat is linear in the weights x_(k), but is highly nonlinear in thelocations t_(k) and therefore cannot be solved using classical linearalgebra. Such a system, however, admits a unique solution when theDiracs locations are distinct, which is obtained by using anannihilating filter method as discussed below.

Call {h_(k)}_(k=0,1, . . . ,K) the filter coefficients with z-transform:

$\begin{matrix}{{H(z)} = {{\sum\limits_{k = 0}^{K}{h_{k}z^{- k}}} = {\prod\limits_{k = 1}^{K}\left( {1 - {u_{k}z^{- 1}}} \right)}}} & (7)\end{matrix}$

That is, the roots of H(z) correspond to the locations u_(k)=e^(−j2πτ)^(k) ^(/τ). It follows that:

$\begin{matrix}\begin{matrix}{{h_{m}*{\hat{x}}_{m}} = {\sum\limits_{k = 0}^{K}{h_{k}{\hat{x}}_{m - k}}}} \\{= {\sum\limits_{k = 0}^{K}{\sum\limits_{k^{\prime} = 1}^{K}{\frac{x_{k^{\prime}}}{\tau}h_{k}u_{k^{\prime}}^{m - k}}}}} \\{= {\sum\limits_{k^{\prime} = 1}^{K}{\frac{x_{k^{\prime}}}{\tau}u_{k^{\prime}}^{m}\underset{\underset{{H{(u_{k^{\prime}})}} = 0}{}}{\sum\limits_{k = 0}^{K}{h_{k}u_{k^{\prime}}^{- k}}}}}} \\{= 0}\end{matrix} & (8)\end{matrix}$

The filter h_(m) may thus be referred to as annihilating filter since itannihilates the discrete signal {circumflex over (x)}_(m). The zeros ofthis filter uniquely define the locations t_(k) of the Diracs. Sinceh₀=1, the filter coefficients h_(m) are found from (8) by involving atleast 2K consecutive values of {circumflex over (x)}_(m), leading to alinear system of equations; e.g., if we have {circumflex over (x)}_(m)for m=−K,−K+1, . . . ,K−1, this system can be written in square Toeplitzmatrix form as follows:

$\begin{matrix}{{\begin{bmatrix}{\hat{x}}_{- 1} & {\hat{x}}_{- 2} & \cdots & {\hat{x}}_{- K} \\{\hat{x}}_{0} & {\hat{x}}_{- 1} & \cdots & {\hat{x}}_{{- K} + 1} \\\vdots & \vdots & ⋰ & \vdots \\{\hat{x}}_{K - 2} & {\hat{x}}_{K - 3} & \cdots & {\hat{x}}_{- 1}\end{bmatrix}\begin{bmatrix}h_{1} \\h_{2} \\\vdots \\h_{K}\end{bmatrix}} = {- \begin{bmatrix}{\hat{x}}_{0} \\{\hat{x}}_{1} \\\vdots \\{\hat{x}}_{K - 1}\end{bmatrix}}} & (9)\end{matrix}$

If the x_(k)'s do not vanish, this K×K system of equations has a uniquesolution because any h_(m) satisfying it is also such that H(u_(k))=0for k=1, 2, . . . K. Given the filter coefficients h_(m), the locationst_(k) are retrieved from the zeros u_(k) of the z-transform in (7). Theweights x_(k) are then obtained by considering, for instance, Kconsecutive Fourier-series coefficients as given in (6). By writing theexpression of these K coefficients in vector form, a Vandermonde systemof equations is provided which yields a unique solution for the weightsx_(k) since the u_(k)'s are distinct. Here, no more than 2K consecutivecoefficients {circumflex over (x)}_(m) may be needed to solve both theToeplitz system (9) and the Vandermonde system. This confirms that theknowledge of only 2K Fourier-series coefficients is sufficient toretrieve x(t).

Next, the Fourier-series coefficients {circumflex over (x)}_(m) arerelated to the actual measurements y_(n). Assume N≧Bτ then, for n=1, 2,. . . , N:

$\begin{matrix}{y_{n} = {{\langle{{x(t)},{\sin \; {c\left( {{Bt} - n} \right)}}}\rangle} = {\sum\limits_{{m} \leq {\lfloor{B_{\tau}/2}\rfloor}}{T\; {\hat{x}}_{m}^{{j2\pi}\; {{mn}/N}}}}}} & (10)\end{matrix}$

Up to a factor NT=τ, this is simply the inverse Discrete FourierTransform (DFT) of a discrete signal bandlimited to [−└Bτ/2┘,└Bτ/2┘] andwhich coincides with {circumflex over (x)}_(m) in this bandwidth. As aconsequence, the discrete Fourier coefficients of y_(n) provide Bτconsecutive coefficients of the Fourier series of x(t) according toEquation 11:

$\begin{matrix}\begin{matrix}{{\hat{y}}_{m} = {\sum\limits_{n = 1}^{N}{y_{n}^{{- {j2\pi}}\; {{mn}/N}}}}} \\{= \left\{ \begin{matrix}{\tau \; {\hat{x}}_{m}} & {{{if}\mspace{14mu} {m}} \leq \left\lfloor {B\; {\tau/2}} \right\rfloor} \\0 & {{{for}\mspace{14mu} {other}\mspace{14mu} m} \in \left\lbrack {{{- N}/2},{N/2}} \right\rbrack}\end{matrix} \right.}\end{matrix} & (11)\end{matrix}$

Summarizing, at least 2K consecutive coefficients {circumflex over(x)}_(m) are needed to use the annihilating filter method, this meansthat Bτ≧2K. Thus, the bandwidth of the sinc-kernel, B, is always largerthan 2K/τ=ρ, the rate of innovation. However, since Bτ is odd, theminimum number of samples per period is actually one sample larger:N≧B_(min)τ=2K+1 which is the next best thing to critical sampling.Moreover, this reconstruction algorithm is fast and does not involve anyiterative procedures. Typically, the only step that depends on thenumber of samples, N, is the computation of the DFT coefficients of thesamples y_(n), which can be implemented in O(N log₂ N) elementaryoperations using the FFT algorithm. All the other steps of the algorithm(in particular, polynomial rooting) depend on K only; i.e., on the rateof innovation ρ.

Equation 8 indicates that any non-trivial filter{h_(k)}_(k=0,1, . . . ,L) where L≧K that has u_(k)=e^(−j2πτ) ^(k) ^(/τ)as zeros will annihilate the Fourier series coefficients of x(t). Theconverse is true: any filter with transfer function H(z) thatannihilates the {circumflex over (x)}_(m) is automatically such thatH(u_(k))=0 for k=1, 2, . . . ,K. Taking Equation 11 into account, thismeans that for such filters:

$\begin{matrix}{{{\sum\limits_{k = 0}^{L}{h_{k}{\hat{y}}_{m - k}}} = 0},{{{for}\mspace{14mu} {all}\mspace{14mu} {m}} \leq \left\lfloor {B\; {\tau/2}} \right\rfloor}} & (12)\end{matrix}$

These linear equations may be expressed using a matrix formalism: let Abe the Toeplitz matrix:

$\begin{matrix}{{A = {{\overset{\overset{L + {1\mspace{14mu} {columns}}}{}}{\left. \begin{bmatrix}{\hat{y}}_{{- M} + L} & {\hat{y}}_{{- M} + L - 1} & \ldots & {\hat{y}}_{- M} \\{\hat{y}}_{{- M} + L + 1} & {\hat{y}}_{{- M} + L} & \ldots & {\hat{y}}_{{- M} + 1} \\{\hat{y}}_{{- M} + L + 2} & {\hat{y}}_{{- M} + L + 1} & ⋰ & \vdots \\\vdots & ⋰ & ⋰ & {\hat{y}}_{{- M} + L} \\\vdots & ⋰ & ⋰ & \vdots \\{\hat{y}}_{M} & {\hat{y}}_{M - 1} & \ldots & {\hat{y}}_{M - L}\end{bmatrix} \right\}}\mspace{14mu} 2M} - L + {1\mspace{14mu} {rows}}}},{{{where}\mspace{14mu} M} = \left\lfloor {B\; {\tau/2}} \right\rfloor}} & (13)\end{matrix}$

and H=[h₀, h₁, . . . , h_(L)]^(T) the vector containing the coefficientsof the annihilating filter, then Equation 12 is equivalent to:

AH=0   (14)

which can be seen as a rectangular extension of Equation 9. Note that,unlike Equation 7, H is not restricted to satisfy h₀=1. Now, if wechoose L>K, there are L−K+1 independent polynomials of degree L withzeros at {u_(k)}_(k=1,2, . . . ,K), which means that there are L−K+1independent vectors H which satisfy Equation 14. As a consequence, therank of the matrix A does never exceed K. This provides a simple way todetermine K when it is not known a priori: find the smallest L such thatthe matrix A built according to (13) is singular, then K=L−1.

The annihilation property of Equation 12 satisfied by the DFTcoefficients ŷ_(m) is narrowly linked to the periodized sinc-Dirichletwindow used prior to sampling. This approach may be generalized to otherkernels such as for example the (non-periodized) sinc, the Gaussianwindows, and any window that satisfies a Strang-Fix like condition,i.e., the reproduction of polynomials.

Finite Rate Innovation (FRI) Signals with Noise

“Noise”, or more generally model mismatch are unfortunately omnipresentin data acquisition, making the solution presented in the previoussection an ideal case. As shown on FIGS. 1 and 2, perturbations to theFRI model may arise both in the analog domain during, e.g., atransmission procedure, and in the digital domain during and aftersampling—in this respect, quantization is a source of noise as well.

According to an aspect of the disclosure, an increase in the samplingrate is performed to achieve robustness against noise.

Consider the signal resulting from the convolution of a τ-periodic FRIsignal (Equation 3) and a sinc-window of bandwidth B, where Bτ is an oddinteger. Due to noise corruption, Equation 4 becomes:

$\begin{matrix}{{y_{n} = {{{\sum\limits_{k = 1}^{K}{x_{k}{\phi \left( {{nT} - t_{k}} \right)}}} + {ɛ_{n}\mspace{14mu} {for}\mspace{14mu} n}} = 1}},2,\ldots \mspace{14mu},N} & (15)\end{matrix}$

where T=τ/N and φ(t) is the Dirichlet kernel (Equation 5). Given thatthe rate of innovation of the signal is ρ, consider N>ρτ samples tofight the perturbation ε_(n), making the data redundant by a factor ofN/(ρτ). Algorithms that may be applied to efficiently exploit this extraredundancy will now be discussed.

One approach utilizes the Total Least-Squares approximation (implementedusing a Singular Value Decomposition), optionally enhanced by an initial“denoising” (or “model matching”) step provided by what is referred toherein as Cadzow's iterated algorithm. The full algorithm, depicted inFIG. 2, is also detailed below in its two main components.

By computing the theoretical minimal uncertainties known as Cramer-Raobounds on the innovation parameters, it may be seen that thesealgorithms exhibit a quasi-optimal behavior down to noise levels of theorder of 5 dB (depending on the number of samples). In particular, thesebounds are instructive in how to choose the bandwidth of the samplingkernel.

Total Least-Squares Method

In the presence of noise, the annihilation Equation 14 is not satisfiedexactly, yet it is still reasonable to expect that the minimization ofthe Euclidian norm ∥AH∥² under the constraint that ∥H∥²=1 may yield aclose estimate of H. Of particular interest is the solution forL=K—annihilating filter of minimal size—because the K zeros of theresulting filter provide a unique estimation of the K locations t_(k).It can be shown that this minimization can be solved by performing asingular value decomposition of A as defined by Equation 13—moreexactly: an eigenvalue decomposition of the matrix A^(T)A—and choosingfor H the eigenvector corresponding to the smallest eigenvalue. Morespecifically, if A=USV^(T) where U is a (Bτ−K)×(K+1) unitary matrix, Sis a (K+1)×(K+1) diagonal matrix with decreasing positive elements, andV is a (K+1)×(K+1) unitary matrix, then H is the last column of V. Oncethe shifts t_(k) are retrieved, the weights x_(k) follow from a leastmean square minimization of the difference between the samples y_(n) andthe FRI model (Equation 15).

We will now describe a possible aspect of the total least square methodfor retrieving the innovations x_(k) and t_(k) from the noisy samples ofEquation 15. The method could comprise following steps:

-   -   1) Compute the N-DFT coefficients of the samples ŷ_(m)=Σ_(n=1)        ^(N)y_(n)e^(−j2πmn/N);    -   2) Choose L=K and build a rectangular Toeplitz matrix A        according to Equation 13;    -   3) Perform a singular value decomposition of the matrix A, and        choose the eigenvector [h₀, h₁, . . . ,h_(k)]^(T) corresponding        to the smallest eigenvalue—i.e., the annihilating filter        coefficients;    -   4) Compute the roots e^(−j2πτ) ^(k) ^(/T) of the z-transform        H(z)=Σ_(k=0) ^(K)h_(k)z^(−k) and deduce {t_(k)}_(k=1, . . . ,K);    -   5) Compute the least mean square solution x_(k) of the N        equations {y_(n)−Σ_(k)x_(k)φ(nT−t_(k))}_(n=1,2, . . . ,N).

Extra Denoising: Cadzow

The total least-square algorithm works quite well for moderate values ofthe noise—a level that depends on the number of Diracs. However, forsmall signal-to-noise ratios (SNR), the results may become unreliableand it is advisable to apply a robust denoising process. In one aspect,the denoising process “projects” the noisy samples onto the sampled FRImodel of Equation 15. It thus replaces the noisy digital samples by anapproximation of the noiseless signal, or by a signal which may still benoisy (less noisy), but which has a finite rate of innovation, or iscloser to a signal with a finite rate of innovation.

In one aspect, when the samples y_(n) are very noisy, it is preferableto first denoise them by performing at least one iteration of Cadzow'salgorithm before applying the above described total least square methodor another method for retrieving the parameters (t_(k), x_(k)) of theother signal.

The Cadzow process delivers a FRI signal, or substantially FRI signal,that can equivalently be represented, either by a sequence of uniformsamples, or by a set of parameters (innovations) that provide thesesamples. The switch between the two representations is performed byusing the Annihilating Filter technique, which may be reduced to the“inverse” operation of the sampling operation.

As discussed above, the noiseless matrix A in Equation 13 is of rank Kwhenever L≧K. The singular value decomposition (SVD) of A may beprovided, where A=USV^(T), and forcing to zero the L+1−K smallestdiagonal coefficients of the matrix S to yield S′. The resulting matrixA′=US′V^(T) is not Toeplitz anymore but its best Toeplitz approximationis obtained by averaging the diagonals of A′. This leads to a new“denoised” sequence ŷ′_(n) that matches the noiseless FRI sample modelbetter than the original ŷ_(n)'s. A few of these iterations lead tosamples that can be expressed almost exactly as samples of an FRIsignal. This FRI signal is the closest to the noiseless one as A iscloser to a square matrix, i.e., L=└Bτ/2┘. The computational cost ofthis algorithm, summarized below, is however higher than the previousannihilating filter method described in relation with noiseless signals,since it requires performing the SVD of a square matrix of large size,typically half the number of samples.

We will now describe a possible aspect of the Cadzow's iterativedenoising method for converting a noisy sequence of samples y_(n) in anoiseless or less noisy sequence y′_(n) corresponding to a FRI, ornearly FRI, signal.

-   -   1) Compute the N-DFT coefficients of the samples ŷ_(m)=Σ_(n=1)        ^(N)y_(n)e^(−j2πmn/N);    -   2) Choose an integer L in [K, Bτ/2] and build the rectangular        Toeplitz matrix A according to Equation 13;    -   3) Perform a singular value decomposition of A=USV^(T) where U        is a (2M−L+1)×(L+1) unitary matrix, S is a diagonal (L+1)×(L+1)        matrix, and V is a (L+1)×(L+1) unitary matrix    -   4) Build the diagonal matrix S′ from S by keeping only the K        most significant diagonal elements, and deduce the total        least-squares approximation of A by A′=US′V^(T);    -   5) Build a denoised approximation ŷ′_(n) of ŷ_(n) by averaging        the diagonals of the matrix A′;    -   6) Iterate step 2 until a condition is met, for example until        the (K+1)^(th) largest diagonal element of S is smaller than the        K^(th) largest diagonal element by a pre-requisite factor        depending on the requested accuracy and/or computing time. The        iterations may also be repeated a predefined number of times, or        until a desired signal to noise ratio of a reconstructed signal        has been reached, or when a desired level of accuracy of the        information relative to the another signal x(t) has been        reached.

Other methods than a singular value decomposition may be used forfinding the coefficients of the annihilating filter, including methodsbased on the solution of a linear system of equation.

For many applications, a small number of iterations, typically less than10, is sufficient. Experimentally, the best choice for L in step 2 isL=M.

Uncertainty Relation for the One-Dirac Case

Let's consider the finite rate of innovation problem of finding [x₁, t₁]from a set of N noisy samples [y₁, y₂, . . . , y_(N)]

y _(n)=μ_(n)+ε_(n) with μ_(n) =x ₁φ(nτ/N−t ₁)   (16)

where φ(t) is the τ-periodic, B-bandlimited Dirichlet kernel and ε_(n)is a stationary Gaussian noise. Any unbiased algorithm that estimates t₁and x₁ will do so up to an error quantified by their standard deviationΔt₁ and Δx₁, lower bounded by Cramér-Rao formulae. Denoting the noisepower by σ² and the Peak signal-to-noise ratio by PSNR=|x₁|²/σ², twocases may be considered:

If the noise is white, i.e. its power spectrum density is constant andequals σ², then:

$\begin{matrix}{{\frac{\Delta \; t_{1}}{\tau} \geq {\frac{1}{\pi}{\sqrt{\frac{3B\; \tau}{N\left( {{B^{2}\tau^{2}} - 1} \right)}} \cdot {PSNR}^{{- 1}/2}}}}\mspace{14mu} {and}{\frac{\Delta \; x_{1}}{x_{1}} \geq {\sqrt{\frac{B\; \tau}{N}} \cdot {PSNR}^{{- 1}/2}}}} & (17)\end{matrix}$

If the noise is a white noise filtered by φ(t), then we find

$\begin{matrix}{{\frac{\Delta \; t_{1}}{\tau} \geq {\frac{1}{\pi}{\sqrt{\frac{3}{{B^{2}\tau^{2}} - 1}} \cdot {PSNR}^{{- 1}/2}}}}\mspace{14mu} {and}\mspace{11mu} \; {\frac{\Delta \; x_{1}}{x_{1}} \geq {PSNR}^{{- 1}/2}}} & (18)\end{matrix}$

In both configurations, we conclude that in order to minimize theuncertainty on t₁, it is better to maximize the bandwidth of theDirichlet kernel, i.e., to choose the bandwidth B of the sampling kernelsuch that Bτ=N if N is odd, or such that Bτ=N−1 if N is even. Since Bτ≦Nwe always have the following uncertainty relation

$\begin{matrix}{{N \cdot {PSNR}^{1/2} \cdot \frac{\Delta \; t_{1}}{\tau}} \geq \frac{\sqrt{3}}{\pi}} & (19)\end{matrix}$

involving the number of measurements N, the end noise level and theuncertainty on the position.

Extra Denoising: Other Than Cadzow

The above described Cadzow denoising process outputs a signal with animproved signal-to-noise ratio, but in which the parameters (for examplethe location of the Diracs and their amplitude) are still noisy. Themain advantage of this process is to deliver a signal which isparametric, i.e., a signal with a Finite Rate of Innovation, or at leasta signal with a substantially Finite Rate of Innovation, or sufficientlyclose for the intended purpose to a signal with a Finite Rate ofInnovation. The Cadzow process is thus not only a denoising process thatmerely improves the SNR ratio, in the way a Wiener filter for examplewould perform, but also projects the signal into the subspace of FRIsignals for which reconstruction methods are available.

However, other denoising process and methods may be used in theframework of the disclosure. As a particular denoising process, wealready mentioned the singular value decomposition that can be used toget a rank K approximation to the noisy matrix, which is a subspaceapproximation. Other subspace techniques, in particular variants basedon the ESPRIT algorithm and/or on wavelets arrays, may be used forreducing the level of noise and for finding parameters of the signal onewants to reconstruct or for which one wants to retrieve information. Inthis case, the denoising and estimation of information relating to theanother signal (x_(t)) may be combined in one common process.

The teachings herein may be incorporated into a device employing variouscomponents for communicating with at least one other device. FIG. 4depicts several sample components that may be employed to facilitatecommunication between devices. Here, a first device 702 and a seconddevice 704 are adapted to communicate via a communication link 706, forexample a wireless communication link, over a suitable medium.

Initially, components involved in sending information from the device702 to the device 704 (e.g., a reverse link) will be treated. A transmit(“TX”) data processor 708 receives traffic data (e.g., data packets)from a data buffer 710 or some other suitable component. The transmitdata processor 708 processes (e.g., encodes, interleaves, and symbolmaps) each data packet based on a selected coding and modulation scheme,and provides data symbols. In general, a data symbol is a modulationsymbol for data, and a pilot symbol is a modulation symbol for a pilot(which is known a priori). A modulator 712 receives the data symbols,pilot symbols, and possibly signaling for the reverse link, and performsmodulation (e.g., OFDM, PPM or some other suitable modulation) and/orother processing as specified by the system, and provides a stream ofoutput chips. A transmitter (“TMTR”) 714 processes (e.g., converts toanalog, filters, amplifies, and frequency upconverts) the output chipstream and generates a modulated signal, which is then transmitted froman antenna 716.

The modulated signals transmitted by the device 702 (along with signalsfrom other devices in communication with the device 704) are received byan antenna 718 of the device 704. Noise may be added during thetransmission over the channel and/or during reception. A receiver(“RCVR”) 720 processes (e.g., conditions and digitizes) the receivedsignal from the antenna 718 and provides received samples. A demodulator(“DEMOD”) 722 processes (e.g., demodulates and detects) the receivedsamples and provides detected data symbols, which may be a noisyestimate of the data symbols transmitted to the device 704 by the otherdevice(s). A receive (“RX”) data processor 724 processes (e.g., symboldemaps, deinterleaves, and decodes) the detected data symbols andprovides decoded data associated with each transmitting device (e.g.,device 702).

Components involved in sending information from the device 704 to thedevice 702 (e.g., a forward link) will be now be treated. At the device704, traffic data is processed by a transmit (“TX”) data processor 726to generate data symbols. A modulator 728 receives the data symbols,pilot symbols, and signaling for the forward link, performs modulation(e.g., OFDM or some other suitable modulation) and/or other pertinentprocessing, and provides an output chip stream, which is furtherconditioned by a transmitter (“TMTR”) 730 and transmitted from theantenna 718. In some implementations signaling for the forward link mayinclude power control commands and other information (e.g., relating toa communication channel) generated by a controller 732 for all devices(e.g. terminals) transmitting on the reverse link to the device 704.

At the device 702, the modulated signal transmitted by the device 704 isreceived together with noise by the antenna 716, conditioned anddigitized by a receiver (“RCVR”) 734, and processed by a demodulator(“DEMOD”) 736 to obtain detected data symbols. A receive (“RX”) dataprocessor 738 processes the detected data symbols and provides decodeddata for the device 702 and the forward link signaling. A controller 740receives power control commands and other information to control datatransmission and to control transmit power on the reverse link to thedevice 704.

The controllers 740 and 732 direct various operations of the device 702and the device 704, respectively. For example, a controller maydetermine an appropriate filter, reporting information about the filter,and decode information using a filter. Data memories 742 and 744 maystore program codes and data used by the controllers 740 and 732,respectively.

FIG. 4 also illustrates that the communication components may includeone or more components that perform operations as taught herein. Forexample, a receive control component 746 may cooperate with thecontroller 740 and/or other components of the device 702 to receiveinformation from another device (e.g., device 704). Similarly, a receivecontrol component 748 may cooperate with the controller 732 and/or othercomponents of the device 704 to receive information from another device(e.g., device 702).

A wireless device may include various components that perform functionsbased on signals that are transmitted by or received at the wirelessdevice or other signals as taught herein. For example, a wirelessheadset may include a transducer arranged to provide an audio outputbased on estimated information, retrieved parameters, or at least onesample as discussed above. A wireless watch may include a user interfacearranged to provide an indication based estimated information, retrievedparameters, or at least one sample as discussed above. A wirelesssensing device may include a sensor arranged to sense (e.g., to providedata to be transmitted) based on estimated information, retrievedparameters, or at least one sample as discussed above (e.g., receivedinformation that controls the sensing). The teachings herein may also beapplied to optical or galvanic transmission channels, including opticaltransmission over optical fibers.

A wireless device may communicate via one or more wireless communicationlinks that are based on or otherwise support any suitable wirelesscommunication technology. For example, in some aspects a wireless devicemay associate with a network. In some aspects the network may comprise abody area network or a personal area network (e.g., an ultra-widebandnetwork). In some aspects the network may comprise a local area networkor a wide area network. A wireless device may support or otherwise useone or more of a variety of wireless communication technologies,protocols, or standards such as, for example, UWB, CDMA, TDMA, OFDM,OFDMA, WiMAX, and Wi-Fi. Similarly, a wireless device may support orotherwise use one or more of a variety of corresponding modulation ormultiplexing schemes. A wireless device may thus include appropriatecomponents (e.g., air interfaces) to establish and communicate via oneor more wireless communication links using the above or other wirelesscommunication technologies. For example, a device may comprise awireless transceiver with associated transmitter and receiver componentsthat may include various components (e.g., signal generators and signalprocessors) that facilitate communication over a wireless medium.

In some aspects a wireless device may communicate via an impulse-basedwireless communication link. For example, an impulse-based wirelesscommunication link may utilize ultra-wideband pulses that have arelatively short length (e.g., on the order of a few nanoseconds orless) and a relatively wide bandwidth. In some aspects theultra-wideband pulses may have a fractional bandwidth on the order ofapproximately 20% or more and/or have a bandwidth on the order ofapproximately 500 MHz or more.

The teachings herein may be incorporated into (e.g., implemented withinor performed by) a variety of apparatuses (e.g., devices). For example,one or more aspects taught herein may be incorporated into a phone(e.g., a cellular phone), a personal data assistant (“PDA”), anentertainment device (e.g., a music or video device), a headset (e.g.,headphones, an earpiece, etc.), a microphone, a medical sensing device(e.g., a biometric sensor, a heart rate monitor, a pedometer, an EKGdevice, a smart bandage, etc.), a user I/O device (e.g., a watch, aremote control, a light switch, a keyboard, a mouse, etc.), anenvironment sensing device (e.g., a tire pressure monitor), a computer,a point-of-sale device, an entertainment device, a hearing aid, aset-top box, a “smart” bandage, or any other suitable device.

The methods and apparatus disclosed in this document are especiallyuseful, without being limited to, the processing of sparse signals, i.e.signals with a low rate of innovation as compared with the bandwidth.

As already mentioned, the methods and apparatus of the application mayalso be used for ultra-wide band (UWB) communications. Thiscommunications method may use pulse position modulation (PPM) with verywideband pulses (up to several gigahertz of bandwidth). Designing adigital receiver using conventional sampling theory would requireanalog-to-digital conversion (ADC) running at very high frequencies, forexample over 5 GHz. Such a receiver would be very expensive in terms ofprice and power consumption. A simple model of an UWB pulse is a Diracconvolved with a wideband, zero mean pulse. At the receiver, the signalis the convolution of the original pulse with the channel impulseresponse, which includes many reflections, and all this buried in highlevels of noise.

More generally, the methods and devices disclosed in this document areespecially useful for sparse signals, i.e. signals with a low rate ofinnovation as compared to the bandwidth, be it in time and/or in space.The methods are however not limited to sparse signals and may be usedfor any signal with a finite rate of innovation, including bandlimitedsignal.

Differently from the methods used by known compressed sensing tools,compressed sensing framework, the methods are not limited to discretevalues; indeed, the innovation times t_(k) and the weights c_(k) usedand retrieved by the methods may assume arbitrary real values.

These devices may have different power and data requirements. In someaspects, the teachings herein may be adapted for use in low powerapplications (e.g., through the use of an impulse-based signaling schemeand low duty cycle modes) and may support a variety of data ratesincluding relatively high data rates (e.g., through the use ofhigh-bandwidth pulses).

In some aspects a wireless device may comprise an access device (e.g., aWi-Fi access point) for a communication system. Such an access devicemay provide, for example, connectivity to another network (e.g., a widearea network such as the Internet or a cellular network) via a wired orwireless communication link. Accordingly, the access device may enableanother device (e.g., a Wi-Fi station) to access the other network orsome other functionality. In addition, it should be appreciated that oneor both of the devices may be portable or, in some cases, relativelynon-portable.

The components described herein may be implemented in a variety of ways.Referring to FIGS. 5A and 5B, apparatuses 800, 900, 1000, and 1100 arerepresented as a series of interrelated functional blocks that mayrepresent functions implemented by, for example, one or more integratedcircuits (e.g., an ASIC) or may be implemented in some other manner astaught herein. As discussed herein, an integrated circuit may include aprocessor, software, other components, or some combination thereof.

The apparatuses 800, 900, 1000, and 1100 may include one or more modulesthat may perform one or more of the functions described above withregard to various figures. For example, an ASIC for obtaining a digitalsignal 802, 902, or 1002 may correspond to, for example, a samplerand/or one or more other components as discussed herein. An ASIC forestimating information 804, 1008, or 1108 may correspond to, forexample, a processor and/or one or more other components as discussedherein. An ASIC for determining a sampling rate 806 or 1118 maycorrespond to, for example, an estimator and/or one or more othercomponents as discussed herein. An ASIC for defining a sampling kernel808 or 1104 may correspond to, for example, an estimator and/or one ormore other components as discussed herein. An ASIC for converting thedigital signal 904 may correspond to, for example, a denoiser and/or oneor more other components as discussed herein. An ASIC for retrievingparameters 906 may correspond to, for example, a processor and/or one ormore other components as discussed herein. An ASIC for building aToeplitz matrix 1004 may correspond to, for example, a processor and/orone or more other components as discussed herein. An ASIC for performingan SVD of the Toeplitz matrix 1006 may correspond to, for example, aprocessor and/or one or more other components as discussed herein. AnASIC for obtaining an analog signal 1102 may correspond to, for example,a receiver and/or one or more other components as discussed herein. AnASIC for using a sampling kernel 1106 may correspond to, for example, asampler and/or one or more other components as discussed herein. An ASICfor defining an annihilating filter 1110 may correspond to, for example,a processor and/or one or more other components as discussed herein. AnASIC for reconstructing 1112 may correspond to, for example, a processorand/or one or more other components as discussed herein. An ASIC forretrieving an estimate 1114 may correspond to, for example, a processorand/or one or more other components as discussed herein. An ASIC forusing a denoising process 1116 may correspond to, for example, adenoiser and/or one or more other components as discussed herein.

As noted above, in some aspects these components may be implemented viaappropriate processor components. These processor components may in someaspects be implemented, at least in part, using structure as taughtherein. In some aspects a processor may be adapted to implement aportion or all of the functionality of one or more of these components.In some aspects one or more of the components, especially componentsrepresented by dashed boxes, are optional.

As noted above, apparatuses 800, 900, 1000, and 1100 may comprise one ormore integrated circuits. For example, in some aspects a singleintegrated circuit may implement the functionality of one or more of theillustrated components, while in other aspects more than one integratedcircuit may implement the functionality of one or more of theillustrated components.

In addition, the components and functions represented by FIGS. 5A and 5Bas well as other components and functions described herein, may beimplemented using any suitable means. Such means also may beimplemented, at least in part, using corresponding structure as taughtherein. For example, the components described above in conjunction withthe “ASIC for” components of FIGS. 5A and 5B also may correspond tosimilarly designated “means for” functionality. Thus, in some aspectsone or more of such means may be implemented using one or more ofprocessor components, integrated circuits, or other suitable structureas taught herein. The methods described and claimed may be carried, atleast in part, using software modules carried out by suitable processingmeans.

Also, it should be understood that any reference to an element hereinusing a designation such as “first,” “second,” and so forth does notgenerally limit the quantity or order of those elements. Rather, thesedesignations may be used herein as a convenient method of distinguishingbetween two or more elements or instances of an element. Thus, areference to first and second elements does not mean that only twoelements may be employed there or that the first element must precedethe second element in some manner. Also, unless stated otherwise a setof elements may comprise one or more elements. In addition, terminologyof the form “at least one of: A, B, or C” used in the description or theclaims means “A or B or C or any combination thereof.”

Those of skill in the art would understand that information and signalsmay be represented using any of a variety of different technologies andtechniques. For example, data, instructions, commands, information,signals, bits, symbols, and chips that may be referenced throughout theabove description may be represented by voltages, currents,electromagnetic waves, magnetic fields or particles, optical fields orparticles, or any combination thereof.

Those of skill would further appreciate that any of the variousillustrative logical blocks, modules, processors, means, circuits, andalgorithm steps described in connection with the aspects disclosedherein may be implemented as electronic hardware (e.g., a digitalimplementation, an analog implementation, or a combination of the two,which may be designed using source coding or some other technique),various forms of program or design code incorporating instructions(which may be referred to herein, for convenience, as “software” or a“software module”), or combinations of both. To clearly illustrate thisinterchangeability of hardware and software, various illustrativecomponents, blocks, modules, circuits, and steps have been describedabove generally in terms of their functionality. Whether suchfunctionality is implemented as hardware or software depends upon theparticular application and design constraints imposed on the overallsystem. Skilled artisans may implement the described functionality invarying ways for each particular application, but such implementationdecisions should not be interpreted as causing a departure from thescope of the present disclosure.

The various illustrative logical blocks, modules, and circuits describedin connection with the aspects disclosed herein may be implementedwithin or performed by an integrated circuit (“IC”), an access terminal,or an access point. The IC may comprise a general purpose processor, adigital signal processor (DSP), an application specific integratedcircuit (ASIC), a field programmable gate array (FPGA) or otherprogrammable logic device, discrete gate or transistor logic, discretehardware components, electrical components, optical components,mechanical components, or any combination thereof designed to performthe functions described herein, and may execute codes or instructionsthat reside within the IC, outside of the IC, or both. A general purposeprocessor may be a microprocessor, but in the alternative, the processormay be any conventional processor, controller, microcontroller, or statemachine. A processor may also be implemented as a combination ofcomputing devices, e.g., a combination of a DSP and a microprocessor, aplurality of microprocessors, one or more microprocessors in conjunctionwith a DSP core, or any other such configuration.

It is understood that any specific order or hierarchy of steps in anydisclosed process is an example of a sample approach. Based upon designpreferences, it is understood that the specific order or hierarchy ofsteps in the processes may be rearranged while remaining within thescope of the present disclosure. The accompanying method claims presentelements of the various steps in a sample order, and are not meant to belimited to the specific order or hierarchy presented.

The steps of a method or algorithm described in connection with theaspects disclosed herein may be embodied directly in hardware, in asoftware module executed by a processor, or in a combination of the two.A software module (e.g., including executable instructions and relateddata) and other data may reside in a data memory such as RAM memory,flash memory, ROM memory, EPROM memory, EEPROM memory, registers, a harddisk, a removable disk, a CD-ROM, or any other form of computer-readablestorage medium known in the art. A sample storage medium may be coupledto a machine such as, for example, a computer/processor (which may bereferred to herein, for convenience, as a “processor”) such theprocessor can read information (e.g., code) from and write informationto the storage medium. A sample storage medium may be integral to theprocessor. The processor and the storage medium may reside in an ASIC.The ASIC may reside in user equipment. In the alternative, the processorand the storage medium may reside as discrete components in userequipment. Moreover, in some aspects any suitable computer-programproduct may comprise a computer-readable medium comprising codes (e.g.,executable by at least one computer) relating to one or more of theaspects of the disclosure. In some aspects a computer program productmay comprise packaging materials.

The previous description of the disclosed aspects is provided to enableany person skilled in the art to make or use the present disclosure.Various modifications to these aspects will be readily apparent to thoseskilled in the art, and the generic principles defined herein may beapplied to other aspects without departing from the scope of thedisclosure. Thus, the present disclosure is not intended to be limitedto the aspects shown herein but is to be accorded the widest scopeconsistent with the principles and novel features disclosed herein.

1. A method of signal processing, comprising: obtaining a digital signal(y_(n)) based on another signal (x_(t)) and noise; and estimatinginformation relating to the another signal (x_(t)) by using a denoisingprocess to produce a denoised signal (y′_(n)) and by processing thedenoised signal (y′_(n)), wherein the denoised signal (y′_(n)) producedby the denoising process has a substantially Finite Rate of Innovation.2. The method of claim 1, wherein processing the denoised signal(y′_(n)) comprises using an annihilating filter or a subspace technique.3. The method of claim 2, wherein the estimation comprises: using thedenoising process to provide the denoised signal (y′_(n)) based on thedigital signal (y_(n)); and using the annihilating filter to operate onthe denoised signal (y′_(n)).
 4. The method of claim 2, furthercomprising defining the annihilating filter by performing a singularvalue decomposition method and minimizing a norm of an associatedmatrix.
 5. The method of claim 2, wherein processing the denoised signal(y′_(n)) comprises reconstructing the another signal (x_(t)) through useof the annihilating filter.
 6. The method of claim 2, wherein processingthe denoised signal (y′_(n)) comprises retrieving an estimate of shifts(t_(k)) associated with the another signal (x_(t)) through use of theannihilating filter.
 7. The method of claim 6, wherein processing thedenoised signal (y′_(n)) comprises retrieving an estimate of weights(x_(k)) associated with the another signal (x_(t)).
 8. The method ofclaim 1, wherein the denoising process uses an annihilating filter. 9.The method of claim 1, wherein the denoising process comprises computinga denoised sequence of samples (y′_(n)).
 10. The method of claim 1,wherein the denoising process is iterative.
 11. The method of claim 10,wherein the iterative denoising process terminates after a definednumber of iterations
 12. The method of claim 10, wherein the iterativedenoising process terminates if a desired signal to noise ratio has beenreached.
 13. The method of claim 10, wherein the iterative denoisingprocess terminates if a desired level of accuracy of the informationrelative to the another signal x(t) has been reached.
 14. The method ofclaim 1, wherein the denoising process comprises an iterative Cadzowprocess.
 15. The method of claim 14, wherein the Cadzow processcomprises: defining a rectangular Toeplitz matrix (A); performing asingular value decomposition (USV^(T)) of the rectangular Toeplitzmatrix (A) to provide a diagonal matrix (S); forcing to zero at leastthe less significant diagonal coefficients of the diagonal matrix (S) toprovide a modified diagonal matrix (S′) and to provide a denoisedapproximation of the rectangular Toeplitz matrix (A′); and providing atleast one denoised sample based on the approximation of the Toeplitzmatrix (A′).
 16. The method of claim 1, wherein the denoising processiteratively uses a singular value decomposition of an approximatedToeplitz matrix (A)
 17. The method of claim 1, wherein obtaining thedigital signal (y_(n)) comprises sampling the another signal (x_(t)) andnoise.
 18. The method of claim 17, wherein the digital signal (y_(n)) isobtained by an apparatus in which a rate for the sampling is definedbefore receipt of the another signal (x_(t)).
 19. The method of claim17, wherein the signal (y_(n)) is obtained by an apparatus thatdetermines rate of innovation or number of innovations of the anothersignal (x_(t)).
 20. The method of claim 17, wherein a sampling rate usedfor the sampling is lower than the minimal sampling rate given by theShannon theorem applied to the another signal (x_(t)).
 21. The method ofclaim 20, wherein the sampling rate is higher than a rate of innovationof the another signal (x_(t)).
 22. The method of claim 21, wherein theanother signal (x_(t)) is not bandlimited.
 23. The method of claim 22,wherein the another signal (x_(t)) has a substantially Finite Rate ofInnovation.
 24. The method of claim 17, further comprising determining asampling rate for the sampling of the another signal (x_(t)) based on anestimate of the noise.
 25. The method of claim 24, wherein the estimatednoise comprises an estimated signal-to-noise ratio.
 26. The method ofclaim 24, wherein the determination of the sampling rate is furtherbased on a desired level of accuracy of the information.
 27. The methodof claim 24, wherein the determination of the sampling rate is furtherbased on a comparison between an assumption of an expected signal tonoise ratio and a signal to noise ratio of a reconstructed signal thatis based on the information.
 28. The method of claim 17, furthercomprising defining a sampling kernel used for the sampling of theanother signal (x_(t)) based on the noise.
 29. The method of claim 28,wherein bandwidth of the sampling kernel is based on the noise.
 30. Themethod of claim 1, wherein the another signal (x_(t)) has a finiteduration and a substantially finite number of innovations during thefinite duration.
 31. The method of claim 1, wherein the another signal(x_(t)) is periodic and has a finite number of innovations during agiven period.
 32. The method of claim 1, wherein the another signal(x_(t)) is a function of at least one of: time or space.
 33. Anapparatus for signal processing, comprising: a circuit arranged toobtain a digital signal (y_(n)) based on another signal (x_(t)) andnoise; and an estimator arranged to estimate information relating to theanother signal (x_(t)), the estimator comprising: a denoiser arranged toproduce a denoised signal (y′_(n)) with a substantially Finite Rate ofInnovation, and a processor arranged to process the denoised signal(y′_(n)).
 34. The apparatus of claim 33, wherein processing the denoisedsignal (y′_(n)) comprises using an annihilating filter or a subspacetechnique.
 35. The apparatus of claim 34, wherein: the denoiser isfurther arranged to provide the denoised signal (y′_(n)) based on thedigital signal (y_(n)); the processor is further arranged to use theannihilating filter to operate on the denoised signal (y′_(n)).
 36. Theapparatus of claim 34, wherein the processor is further arranged todefine the annihilating filter by performing a singular valuedecomposition method and minimizing a norm of an associated matrix. 37.The apparatus of claim 34, wherein the processor is further arranged toreconstruct the another signal (x_(t)) through use of the annihilatingfilter.
 38. The apparatus of claim 34, wherein the processor is furtherarranged to retrieve an estimate of shifts (t_(k)) associated with theanother signal (x_(t)) through use of the annihilating filter.
 39. Theapparatus of claim 38, wherein the processor is further arranged toretrieve an estimate of weights (x_(k)) associated with the anothersignal (x_(t)).
 40. The apparatus of claim 33, wherein the denoiser isfurther arranged to use an annihilating filter.
 41. The apparatus ofclaim 33, wherein the denoiser is further arranged to compute a denoisedsequence of samples (y′_(n)).
 42. The apparatus of claim 33, wherein thedenoiser is further arranged to carry out an iterative denoisingprocess.
 43. The apparatus of claim 42, wherein the iterative denoisingprocess terminates after a defined number of iterations.
 44. Theapparatus of claim 42, wherein the iterative denoising processterminates if a desired signal to noise ratio has been reached.
 45. Theapparatus of claim 42, wherein the iterative denoising processterminates if a desired level of accuracy of the information relative tothe another signal x(t) has been reached.
 46. The apparatus of claim 33,wherein the denoiser is further arranged to carry out an iterativeCadzow process.
 47. The apparatus of claim 46, wherein the Cadzowprocess comprises: defining a rectangular Toeplitz matrix (A);performing a singular value decomposition (USV^(T)) of the rectangularToeplitz matrix (A) to provide a diagonal matrix (S); forcing to zero atleast the less significant diagonal coefficients of the diagonal matrix(S) to provide a modified diagonal matrix (S′) and to provide a denoisedapproximation of the rectangular Toeplitz matrix (A′); and providing atleast one denoised sample based on the approximation of the Toeplitzmatrix (A′).
 48. The apparatus of claim 33, wherein the denoiser isfurther arranged to iteratively use a singular value decomposition of anapproximated Toeplitz Matrix (A)
 49. The apparatus of claim 33, whereinthe circuit further comprises a sampler arranged to sample the anothersignal (x_(t)) and noise to obtain the digital signal (y_(n)).
 50. Theapparatus of claim 49, wherein a rate for the sampling is defined beforereceipt of the another signal (x_(t)).
 51. The apparatus of claim 49,wherein the estimator is further arranged to determine rate ofinnovation or number of innovations of the another signal (x_(t)). 52.The apparatus of claim 49, wherein a sampling rate used for the samplingis lower than the minimal sampling rate given by the Shannon theoremapplied to the another signal (x_(t)).
 53. The apparatus of claim 52,wherein the sampling rate is higher than a rate of innovation of theanother signal (x_(t)).
 54. The apparatus of claim 53, wherein theanother signal (x_(t)) is not bandlimited.
 55. The apparatus of claim54, wherein the another signal (x_(t)) has a substantially Finite Rateof Innovation.
 56. The apparatus of claim 49, arranged to determine asampling rate for the sampling of the another signal (x_(t)) based on anestimate of the noise.
 57. The apparatus of claim 56, wherein theestimated noise comprises an estimated signal-to-noise ratio.
 58. Theapparatus of claim 56, wherein the determination of the sampling rate isfurther based on a desired level of accuracy of the information.
 59. Theapparatus of claim 56, wherein the determination of the sampling rate isfurther based on a comparison between an assumption of an expectedsignal to noise ratio and a signal to noise ratio of a reconstructedsignal that is based on the information.
 60. The apparatus of claim 49,wherein a sampling kernel used for the sampling of the another signal(x_(t)) is defined based on the noise.
 61. The apparatus of claim 60,wherein bandwidth of the sampling kernel is based on the noise.
 62. Theapparatus of claim 33, wherein the another signal (x_(t)) has a finiteduration and a substantially finite number of innovations during thefinite duration.
 63. The apparatus of claim 33, wherein the anothersignal (x_(t)) is periodic and has a finite number of innovations duringa given period.
 64. The apparatus of claim 33, wherein the anothersignal (x_(t)) is a function of at least one of: time or space.
 65. Anapparatus for signal processing, comprising: means for obtaining adigital signal (y_(n)) based on another signal (x_(t)) and noise; andmeans for estimating information relating to the another signal (x_(t))by using a denoising process to produce a denoised signal (y′_(n)) andby processing the denoised signal (y′_(n)), wherein the denoised signal(y′_(n)) produced by the denoising process has a substantially FiniteRate of Innovation.
 66. The apparatus of claim 65, wherein processingthe denoised signal (y′_(n)) comprises using an annihilating filter or asubspace technique.
 67. The apparatus of claim 66, wherein theestimation comprises: using the denoising process to provide thedenoised signal (y′_(n)) based on the digital signal (y_(n)); and usingthe annihilating filter to operate on the denoised signal (y′_(n)). 68.The apparatus of claim 66, wherein the means for estimating defines theannihilating filter by performing a singular value decomposition methodand minimizing a norm of an associated matrix.
 69. The apparatus ofclaim 66, wherein processing the denoised signal comprisesreconstructing the another signal (x_(t)) through use of theannihilating filter.
 70. The apparatus of claim 66, wherein processingthe denoised signal comprises retrieving an estimate of shifts (t_(k))associated with the another signal (x_(t)) through use of theannihilating filter.
 71. The apparatus of claim 70, wherein processingthe denoised signal comprises retrieving an estimate of weights (x_(k))associated with the another signal (x_(t)).
 72. The apparatus of claim65, wherein the denoising process uses an annihilating filter.
 73. Theapparatus of claim 65, wherein the denoising process comprises computinga denoised sequence of samples (y′_(n)).
 74. The apparatus of claim 65,wherein the denoising process is iterative.
 75. The apparatus of claim74, wherein the iterative denoising process terminates after a definednumber of iterations.
 76. The apparatus of claim 74, wherein theiterative denoising process terminates if a desired signal to noiseratio has been reached.
 77. The apparatus of claim 74, wherein theiterative denoising process terminates if a desired level of accuracy ofthe information relative to the another signal x(t) has been reached.78. The apparatus of claim 65, wherein the denoising process comprisesan iterative Cadzow process.
 79. The apparatus of claim 78, wherein theCadzow process comprises: defining a rectangular Toeplitz matrix (A);performing a singular value decomposition (USV^(T)) of the rectangularToeplitz matrix (A) to provide a diagonal matrix (S); forcing to zero atleast the less significant diagonal coefficients of the diagonal matrix(S) to provide a modified diagonal matrix (S′) and to provide a denoisedapproximation of the rectangular Toeplitz matrix (A′); and providing atleast one denoised sample based on the approximation of the Toeplitzmatrix (A′).
 80. The apparatus of claim 65, wherein the denoisingprocess iteratively uses a singular value decomposition of anapproximated Toeplitz Matrix (A)
 81. The apparatus of claim 65, whereinthe means for obtaining samples the another signal (x_(t)) and noise toobtain the digital signal (y_(n)).
 82. The apparatus of claim 81,wherein a rate for the sampling is defined before receipt of the anothersignal (x_(t)).
 83. The apparatus of claim 81, wherein the means forestimating determines rate of innovation or number of innovations of theanother signal (x_(t)).
 84. The apparatus of claim 81, wherein asampling rate used for the sampling is lower than the minimal samplingrate given by the Shannon theorem applied to the another signal (x_(t)).85. The apparatus of claim 84, wherein the sampling rate is higher thana rate of innovation of the another signal (x_(t)).
 86. The apparatus ofclaim 85, wherein the another signal (x_(t)) is not bandlimited.
 87. Theapparatus of claim 86, wherein the another signal (x_(t)) has asubstantially Finite Rate of Innovation.
 88. The apparatus of claim 81,further comprising means for determining a sampling rate for thesampling of the another signal (x_(t)) based on an estimate of thenoise.
 89. The apparatus of claim 88, wherein the estimated noisecomprises an estimated signal-to-noise ratio.
 90. The apparatus of claim88, wherein the determination of the sampling rate is further based on adesired level of accuracy of the information.
 91. The apparatus of claim88, wherein the determination of the sampling rate is further based on acomparison between an assumption of an expected signal to noise ratioand a signal to noise ratio of a reconstructed signal that is based onthe information.
 92. The apparatus of claim 81, further comprising meansfor defining a sampling kernel used for the sampling of the anothersignal (x_(t)) based on the noise.
 93. The apparatus of claim 92,wherein bandwidth of the sampling kernel is based on the noise.
 94. Theapparatus of claim 65, wherein the another signal (x_(t)) has a finiteduration and a substantially finite number of innovations during thefinite duration.
 95. The apparatus of claim 65, wherein the anothersignal (x_(t)) is periodic and has a finite number of innovations duringa given period.
 96. The apparatus of claim 65, wherein the anothersignal (x_(t)) is a function of at least one of: time or space.
 97. Acomputer-program product for signal processing, comprising:computer-readable medium comprising codes executable to: obtain adigital signal (y_(n)) based on another signal (x_(t)) and noise; andestimate information relating to the another signal (x_(t)) by using adenoising process to produce a denoised signal (y′_(n)) and byprocessing the denoised signal (y′_(n)), wherein the denoised signal(y′_(n)) produced by the denoising process has a substantially FiniteRate of Innovation.
 98. A headset, comprising: a circuit arranged toobtain a digital signal (y_(n)) based on another signal (x_(t)) andnoise; an estimator arranged to estimate information relating to theanother signal (x_(t)), the estimator comprising: a denoiser arranged toproduce a denoised signal (y′_(n)) with a substantially Finite Rate ofInnovation, and a processor arranged to process the denoised signal(y′_(n)); and a transducer arranged to provide an audio output based onthe information.
 99. A watch, comprising: a circuit arranged to obtain adigital signal (y_(n)) based on another signal (x_(t)) and noise; anestimator arranged to estimate information relating to the anothersignal (x_(t)), the estimator comprising: a denoiser arranged to producea denoised signal (y′_(n)) with a substantially Finite Rate ofInnovation, and a processor arranged to process the denoised signal(y′_(n)); and a user interface arranged to provide an indication basedon the information.
 100. A sensing device, comprising: a circuitarranged to obtain a digital signal (y_(n)) based on another signal(x_(t)) and noise; an estimator arranged to estimate informationrelating to the another signal (x_(t)), the estimator comprising: adenoiser arranged to produce a denoised signal (y′_(n)) with asubstantially Finite Rate of Innovation, and a processor arranged toprocess the denoised signal (y′_(n)); and a sensor arranged to sensebased on the information.
 101. A method of signal processing,comprising: obtaining a digital signal (y_(n)) based at least on anothersignal (x_(t)); converting the digital signal (y_(n)) into a converteddigital signal with a substantially Finite Rate of Innovation; andretrieving parameters of the another signal (x_(t)) based on theconverted digital signal.
 102. The method of claim 101, whereinobtaining the digital signal (y_(n)) comprises sampling, at a samplingrate, a composite signal comprising the another signal and noise, thesampling rate being lower than the minimal frequency given by theShannon theorem applied to the composite signal, and the sampling ratebeing higher than a rate of innovation of the another signal (x_(t)).103. The method of claim 102, wherein the composite signal has a nonFinite Rate of Innovation.
 104. The method of claim 101, whereinconversion of the digital signal (y_(n)) is based on an iterative Cadzowprocess.
 105. An apparatus for signal processing, comprising: a circuitarranged to obtain a digital signal (y_(n)) based at least on anothersignal (x_(t)); and an estimator arranged to convert the digital signal(y_(n)) into a converted digital signal with a substantially Finite Rateof Innovation, and further arranged to retrieve parameters of theanother signal (x_(t)) based on the converted digital signal.
 106. Theapparatus of claim 105, wherein the circuit comprises a sampler arrangedto sample, at a sampling rate, a composite signal comprising the anothersignal (x_(t)) and noise, the sampling rate being lower than the minimalfrequency given by the Shannon theorem applied to the composite signal,and the sampling rate being higher than a rate of innovation of theanother signal (x_(t)).
 107. The apparatus of claim 106, wherein thecomposite signal has a non Finite Rate of Innovation.
 108. The apparatusof claim 105, wherein the conversion of the digital signal (y_(n)) isbased on an iterative Cadzow process.
 109. An apparatus for signalprocessing, comprising: means for obtaining a digital signal (y_(n))based at least on another signal (x_(t)); means for converting thedigital signal (y_(n)) into a converted digital signal with asubstantially Finite Rate of Innovation; and means for retrievingparameters of the another signal (x_(t)) based on the converted digitalsignal.
 110. The apparatus of claim 109, wherein obtaining the digitalsignal (y_(n)) comprises sampling, at a sampling rate, a compositesignal comprising the another signal and noise, the sampling rate beinglower than the minimal frequency given by the Shannon theorem applied tothe composite signal, and the sampling rate being higher than a rate ofinnovation of the another signal (x_(t)).
 111. The apparatus of claim110, wherein the composite signal has a non Finite Rate of Innovation.112. The apparatus of claim 109, wherein conversion of the digitalsignal (y_(n)) is based on an iterative Cadzow process.
 113. Acomputer-program product for signal processing, comprising:computer-readable medium comprising codes executable to: obtain adigital signal (y_(n)) based at least on another signal (x_(t)); convertthe digital signal (y_(n)) into a converted digital signal with asubstantially Finite Rate of Innovation; and retrieve parameters of theanother signal (x_(t)) based on the converted digital signal.
 114. Aheadset, comprising: a circuit arranged to obtain a digital signal(y_(n)) based at least on another signal (x_(t)); an estimator arrangedto convert the digital signal (y_(n)) into a converted digital signalwith a substantially Finite Rate of Innovation, and further arranged toretrieve parameters of the another signal (x_(t)) based on the converteddigital signal; and a transducer arranged to provide an audio outputbased on the retrieved parameters.
 115. A watch, comprising: a circuitarranged to obtain a digital signal (y_(n)) based at least on anothersignal (x_(t)); an estimator arranged to convert the digital signal(y_(n)) into a converted digital signal with a substantially Finite Rateof Innovation, and further arranged to retrieve parameters of theanother signal (x_(t)) based on the converted digital signal; and a userinterface arranged to provide an indication based on the retrievedparameters.
 116. A sensing device, comprising: a circuit arranged toobtain a digital signal (y_(n)) based at least on another signal(x_(t)); an estimator arranged to convert the digital signal (y_(n))into a converted digital signal with a substantially Finite Rate ofInnovation, and further arranged to retrieve parameters of the anothersignal (x_(t)) based on the converted digital signal; and a sensorarranged to sense based on the retrieved parameters.
 117. A method ofsignal processing, comprising: obtaining a digital signal (y_(n)) basedon another signal (x_(t)) and noise; building a Toeplitz matrix (A)based on the digital signal (y_(n)); performing a singular valuedecomposition of the Toeplitz matrix (A), and retrieving an eigenvectorcorresponding to a smallest eigenvalue; and estimating informationrelating to the another signal (x_(t)) based on the eigenvector.
 118. Anapparatus for signal processing, comprising: a circuit arranged toobtain a digital signal (y_(n)) based on another signal (x_(t)) andnoise; and an estimator arranged to: build a Toeplitz matrix (A) basedon the digital signal (y_(n)); perform a singular value decomposition ofthe Toeplitz matrix (A), and retrieve an eigenvector corresponding to asmallest eigenvalue; and estimate information relating to the anothersignal (x_(t)) based on the eigenvector.
 119. An apparatus for signalprocessing, comprising: means for obtaining a digital signal (y_(n))based on another signal (x_(t)) and noise; means for building a Toeplitzmatrix (A) based on the digital signal (y_(n)); means for performing asingular value decomposition of the Toeplitz matrix (A), and retrievingan eigenvector corresponding to a smallest eigenvalue; and means forestimating information relating to the another signal (x_(t)) based onthe eigenvector.
 120. A computer-program product for signal processing,comprising: computer-readable medium comprising codes executable to:obtain a digital signal (y_(n)) based on another signal (x_(t)) andnoise; build a Toeplitz matrix (A) based on the digital signal (y_(n));perform a singular value decomposition of the Toeplitz matrix (A), andretrieving an eigenvector corresponding to a smallest eigenvalue; andestimate information relating to the another signal (x_(t)) based on theeigenvector.
 121. A headset, comprising: a circuit arranged to obtain adigital signal (y_(n)) based on another signal (x_(t)) and noise; anestimator arranged to: build a Toeplitz matrix (A) based on the digitalsignal (y_(n)); perform a singular value decomposition of the Toeplitzmatrix (A), and retrieve an eigenvector corresponding to a smallesteigenvalue; and estimate information relating to the another signal(x_(t)) based on the eigenvector; and a transducer arranged to providean audio output based on the information.
 122. A watch, comprising: acircuit arranged to obtain a digital signal (y_(n)) based on anothersignal (x_(t)) and noise; an estimator arranged to: build a Toeplitzmatrix (A) based on the digital signal (y_(n)); perform a singular valuedecomposition of the Toeplitz matrix (A), and retrieve an eigenvectorcorresponding to a smallest eigenvalue; and estimate informationrelating to the another signal (x_(t)) based on the eigenvector; and auser interface arranged to provide an indication based on theinformation.
 123. A sensing device, comprising: a circuit arranged toobtain a digital signal (y_(n)) based on another signal (x_(t)) andnoise; an estimator arranged to: build a Toeplitz matrix (A) based onthe digital signal (y_(n)); perform a singular value decomposition ofthe Toeplitz matrix (A), and retrieve an eigenvector corresponding to asmallest eigenvalue; and estimate information relating to the anothersignal (x_(t)) based on the eigenvector; and a sensor arranged to sensebased on the information.